The Fibonacci sequence was invented by the Italian Leonardo Pisano Bigollo , who is known in mathematical history by several names: Leonardo of Pisa Pisano means "from Pisa" and Fibonacci which means "son of Bonacci".
Fibonacci, the son of an Italian businessman from the city of Pisa, grew up in a trading colony in North Africa during the Middle Ages. Italians were some of the western world's most proficient traders and merchants during the Middle Ages, and they needed arithmetic to keep track of their commercial transactions.
While growing up in North Africa, Fibonacci learned the more efficient Hindu-Arabic system of arithmetical. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair one male, one female every month from the second month on. The puzzle that Fibonacci posed was How many pairs will there be in one year?
At the end of the first month, they mate, but there is still only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits. At the end of the third month, the original female produces a second pair, making 3 pairs in all.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produced her first pair also, making 5 pairs. Now imagine that there are pairs of rabbits after months. The number of pairs in month will be in this problem, rabbits never die plus the number of new pairs born.
But new pairs are only born to pairs at least 1 month old, so there will be new pairs. So we have which is simply the rule for generating the Fibonacci numbers: add the last two to get the next.
Following this through you'll find that after 12 months or 1 year , there will be pairs of rabbits. Bees are better The rabbit problem is obviously very contrived, but the Fibonacci sequence does occur in real populations.
Honeybees provide an example. In a colony of honeybees there is one special female called the queen. The other females are worker bees who, unlike the queen bee, produce no eggs.
The male bees do no work and are called drone bees. Males are produced by the queen's unfertilised eggs, so male bees only have a mother but no father. All the females are produced when the queen has mated with a male and so have two parents.
Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home a hive in search of a place to build a new nest. So female bees have two parents, a male and a female whereas male bees have just one parent, a female.
He has 1 parent, a female. He has 2 grandparents, since his mother had two parents, a male and a female. He has 3 great-grandparents: his grandmother had two parents but his grandfather had only one. How many great-great-grandparents did he have? Again we see the Fibonacci numbers :.
Bee populations aren't the only place in nature where Fibonacci numbers occur, they also appear in the beautiful shapes of shells. To see this, let's build up a picture starting with two small squares of size 1 next to each other. We can now draw a new square — touching both one of the unit squares and the latest square of side 2 — so having sides 3 units long; and then another touching both the 2-square and the 3-square which has sides of 5 units.
We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles. If we now draw a quarter of a circle in each square, we can build up a sort of spiral.
The spiral is not a true mathematical spiral since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals called logarithmic spirals are seen in the shape of shells of snails and sea shells. The image below of a cross-section of a nautilus shell shows the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows.
The chambers provide buoyancy in the water. Fibonacci numbers also appear in plants and flowers. Some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! A particularly beautiful appearance of fibonacci numbers is in the spirals of seeds in a seed head.
The next time you see a sunflower, look at the arrangements of the seeds at its centre. They appear to be spiralling outwards both to the left and the right. At the edge of this picture of a sunflower, if you count those curves of seeds spiralling to the left as you go outwards, there are 55 spirals.
At the same point there are 34 spirals of seeds spiralling to the right. A little further towards the centre and you can count 34 spirals to the left and 21 spirals to the right. The pair of numbers counting spirals curving left and curving right are almost always neighbours in the Fibonacci series.
The same happens in many seed and flower heads in nature. The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges. Nature seems to use the same pattern to arrange petals around the edge of a flower and to place leaves round a stem.
What is more, all of these maintain their efficiency as the plant continues to grow and that's a lot to ask of a single process! So just how do plants grow to maintain this optimality of design? Botanists have shown that plants grow from a single tiny group of cells right at the tip of any growing plant, called the meristem. There is a separate meristem at the end of each branch or twig where new cells are formed. Once formed, they grow in size, but new cells are only formed at such growing points.
Cells earlier down the stem expand and so the growing point rises. Also, these cells grow in a spiral fashion: it's as if the meristem turns by an angle, produces a new cell, turns again by the same angle, produces a new cell, and so on. These cells may then become a seed, a new leaf, a new branch, or perhaps on a flower become petals and stamens. The leaves here are numbered in turn — each is exactly 0.
The amazing thing is that a single fixed angle of rotation can produce the optimal design no matter how big the plant grows. Making 0. But where does this magic number 0. If we take the ratio of two successive numbers in Fibonacci's series, dividing each by the number before it, we will find the following series of numbers:.
If you plot a graph of these values you'll see that they seem to be tending to a limit, which we call the golden ratio also known as the golden number and golden section.
It has a value of approximately 1. The closely related value which we write as , a lowercase phi, is just the decimal part of Phi, namely 0. But why do we see phi in so many plants? The number Phi 1. After two turns through half of a circle we would be back to where the first seed was produced. Over time, turning by half a turn between seeds would produce a seed head with two arms radiating from a central point, leaving lots of wasted space.
A seed head produced by 0. Something similar happens for any other simple fraction of a turn: seeds grow in spiral arms that leave a lot of space between them the number of arms is the denominator of the fraction. So the best value for the turns between seeds will be an irrational number. But not just any irrational number will do. Fibonacci ended his travels around the year and at that time he returned to Pisa.
There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made.
Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works.
However, we know that he wrote some other texts which, unfortunately, are lost. One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance.
Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.
Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around These scholars included Michael Scotus who was the court astrologer, Theodorus Physicus the court philosopher and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci.
We give some details of one of these problems below. After there is only one known document which refers to Fibonacci.
0コメント